Armature is only affected by one bone
Solid objects can be parented directly to bones sans modifier.
In addition to @moonboot's answer am going to suggest a different route via separate objects and bone parenting, avoiding the need for the armature modifier and associated bone groups.
Since it is probably unlikely the &quot;bone&quot; objects are going to be deformed by the armature bones can parent directly to bones.
First step IMO would be to decimate the vert count, without a powerful computer the amount of verts in mesh will tend to make response slow.
Step 1.
Remove the armature modifier.
Make the bones all separate objects. For example sake just skull and jaw, but all teeth / eye-bones?? could be separate objects and parented accordingly. (to object or bone)

Selected all verts of skull P Selection to make the selection a different object. Now have an object each for skull and jaw.
Step 2.
In this example can see no reason to have a left and right jaw bone, so have merged into a sole jaw bone.

EDIT.

The jaw bone is halved to allow for a specific animation where the jaw
splits apart and fires a beam of energy between them.

In which case would keep the sole jaw bone, and make the two half bones children of it, each half of jaw object parented to the half bones.  In which case only need to animate jaw bone for regular mouth open close.

Notice the skull object is already parented to the skull bone.

Select the jaw object, then Shift select the armature and make the jaw bone active. CtrlP Bone. The jaw object is now parented to jaw bone.
Step 3
Parented the jaw bone to the skull bone. (Sounds like a song.) so if the skull is rotated the jaw comes along.

The jaw can be rotated relative to the skull.